Question 1163196
this can be solved geometrically, but you can also solve it using trig identities.


the trig identity you can use is:


cos(a - b) = cos(a) * cos(b) + sin(a) * sin(b)


the equation you want to prove is true is:


cos(pi/2) - x) = sin(x)


let a = pi/2 and b = x


the identity of:


cos(a - b) = cos(a) * cos(b) + sin(a) * sin(b) becomes:


cos(pi/2 - x) = cos(pi/2) * cos(x) + sin(pi/2) * sin(x)


cos(pi/2) = 0
sin(pi/2) = 1


cos(pi/2 - x) = cos(pi/2) * cos(x) + sin(pi/2) * sin(x) becomes:
cos(pi/2 - x) = 0 * cos(x) + 1 * sin(x)
simplify  to get:
cos(pi/2 - x) = sin(x)


there's your proof.


a nice list of trigonometric identities can be found at:
<a href = "https://www.purplemath.com/modules/idents.htm" target = "_blank">https://www.purplemath.com/modules/idents.htm</a>


the identity you are looking for will be under the title of angle sum and difference identities.


you can also solve this geometrically as follows:


draw a right triangle ABC with the right angle at C.
since the sum of the angles of a triangle = 180, then:
angle A + angle B + angle C = 180
since angle C = 90 degrees, then:
angle A + angle B + 90 degrees = 180 degrees
subtract 90 from both sides of that equation to get:
angle A + angle B = 90 degrees.
solve for angle B to get:
angle B = 90 - angle A.
in the triangle ABC, the side opposite angle A is side a, the side opposite angle B is side b, the side opposite angle C is side c.
since C is the 90 degree angle, then the hypotenuse of the triangle is side c.
cos(angle B) = adjacent / hypotenuse = side a divided by side c
sin(angle A) = opposite / hypotenuse = side a divided by side c
this makes cos(angle B) = sin(angle A)
since angle B = 90 - angle A, this makes cos(90 - angle A) = sin(angle A)
let angle A = x, then this equation becomes:
cos(90 - x) = sin(x)
translate 90 degrees to radians to get:
90 degrees * pi / 180 = pi/2 radians.
replace 90 degrees with pi/2 radians to get cos(pi/2 - x) = sin(x)
this also proves the equation is an identity.


here's a diagram that helps you to see what's happening with the geometric proof.


<img src = "http://theo.x10hosting.com/2020/080701.jpg" >